It is relatively easy to study \(I_{\alpha }\)  when \(y\)  is not too close to \(x\)  and not too small, using the saddle point method for friable numbers as developed by Hildebrand and Tenenbaum [1]}.
Let \(\alpha =\alpha (x,y)\)  be defined as the unique positive solution to the following equation:
\(\log x = \sum _{p \le y} \frac{\log p}{p^{\alpha }-1}.\) 
